Problem: $f(x, y) = e^{xy}$ $\dfrac{\partial^2 f}{\partial y^2} = $
Solution: Taking a second order partial derivative is like taking a regular second order derivative. We take the partial derivative once, then we take another partial derivative. $\dfrac{\partial^2 f}{\partial y^2} = \dfrac{\partial}{\partial y} \left[ \dfrac{\partial f}{\partial y} \right]$ Let's differentiate! $\begin{aligned} \dfrac{\partial^2 f}{\partial y^2} &= \dfrac{\partial}{\partial y} \left[ \dfrac{\partial}{\partial y} \left[ e^{xy} \right] \right] \\ \\ &= \dfrac{\partial}{\partial y} \left[ x e^{xy} \right] \\ \\ &= x^2 e^{xy} \end{aligned}$ Therefore, $\dfrac{\partial^2 f}{\partial y^2} = x^2 e^{xy}$.